Análisis de Sistemas Discretos en el Dominio del Tiempo - Convolución Circular y el DFT

""<DIV class=para id=p1_intro><FONT size=2>Usted debería familiarizarse con la </FONT><FONT size=2>convolución discreta</FONT><FONT size=2>, que nos explica como dos señales discretas <SPAN class=mrow id=id7013553><SPAN class=mrow id=id7013557><SPAN class=mrow id=id7013565><SPAN class=mi1>x</SPAN><SPAN class=mo id=id7013568></SPAN><SPAN id=id7013570L>[</SPAN><SPAN id=id7013570M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7013570R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7013570M.offsetHeight; mrowStretch(id7013570L,'é','ê','ê','ë'); mrowStretch(id7013570R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>, la entrada del sistema, y <SPAN class=mrow id=id7013596><SPAN class=mrow id=id7013597><SPAN class=mrow id=id7013602><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7013605></SPAN><SPAN id=id7013607L>[</SPAN><SPAN id=id7013607M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7013607R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7013607M.offsetHeight; mrowStretch(id7013607L,'é','ê','ê','ë'); mrowStretch(id7013607R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>, la respuesta del sistema, se puede definir el resultado del sistema como </FONT> <DIV class=equation id=eq_csum><SPAN class=mrow id=id7013638><SPAN class=mrow id=id7013640> <TABLE class=mtable> <TBODY> <TR> <TD><SPAN class=mrow id=id7013661><SPAN class=mrow id=id7013664><FONT size=2><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7013667></SPAN><SPAN id=id7013668L>[</SPAN><SPAN id=id7013668M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7013668R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7013668M.offsetHeight; mrowStretch(id7013668L,'é','ê','ê','ë'); mrowStretch(id7013668R,'ù','ú','ú','û');</SCRIPT> </FONT></SPAN></SPAN></TD> <TD><SPAN class=mo id=id7013678><FONT size=2>=</FONT></SPAN></TD> <TD><SPAN class=mrow id=id7013684><SPAN class=mrow id=id7013686><FONT size=2><SPAN class=mrow id=id7013688><SPAN class=mi1>x</SPAN><SPAN class=mo id=id7013692></SPAN><SPAN id=id7013694L>[</SPAN><SPAN id=id7013694M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7013694R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7013694M.offsetHeight; mrowStretch(id7013694L,'é','ê','ê','ë'); mrowStretch(id7013694R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7013701>*</SPAN><SPAN class=mrow id=id7013703><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7013706></SPAN><SPAN id=id7013707L>[</SPAN><SPAN id=id7013707M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7013707R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7013707M.offsetHeight; mrowStretch(id7013707L,'é','ê','ê','ë'); mrowStretch(id7013707R,'ù','ú','ú','û');</SCRIPT> </SPAN></FONT></SPAN></SPAN></TD></TR> <TR> <TD><FONT size=2></FONT></TD> <TD><SPAN class=mo id=id7013721><FONT size=2>=</FONT></SPAN></TD> <TD><SPAN class=mrow id=id7013727><SPAN class=mrow id=id7013728> <TABLE class=munderover> <TBODY> <TR> <TD><SPAN class=mrow id=id7013745><SPAN class=mi1><FONT size=2>∞</FONT></SPAN></SPAN></TD></TR> <TR> <TD><SPAN class=mo id=id7013730><FONT size=2>∑</FONT></SPAN></TD></TR> <TR> <TD><SPAN class=mrow id=id7013733><FONT size=2><SPAN class=mi1>k</SPAN><SPAN class=mo id=id7013737>=</SPAN><SPAN class=mrow id=id7013740><SPAN class=mo id=id7013741>−</SPAN><SPAN class=mi1>∞</SPAN></SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><FONT size=2><SPAN id=id7013749L>(</SPAN><SPAN id=id7013749M><SPAN class=mrow id=id7013752><SPAN class=mrow id=id7013755><SPAN class=mi1>x</SPAN><SPAN class=mo id=id7013758></SPAN><SPAN id=id7013760L>[</SPAN><SPAN id=id7013760M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7013760R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7013760M.offsetHeight; mrowStretch(id7013760L,'é','ê','ê','ë'); mrowStretch(id7013760R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7013767></SPAN><SPAN class=mrow id=id7013768><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7013772></SPAN><SPAN id=id7013773L>[</SPAN><SPAN id=id7013773M><SPAN class=mrow id=id7013780><SPAN class=mi1>n</SPAN><SPAN class=mo id=id7013783>−</SPAN><SPAN class=mi1>k</SPAN></SPAN></SPAN><SPAN id=id7013773R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7013773M.offsetHeight; mrowStretch(id7013773L,'é','ê','ê','ë'); mrowStretch(id7013773R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN><SPAN id=id7013749R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7013749M.offsetHeight; mrowStretch(id7013749L,'æ','ç','ç','è'); mrowStretch(id7013749R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript></SCRIPT> </SPAN></SPAN><FONT size=2><SPAN class=equation-number>(1)</SPAN> </FONT></DIV><FONT size=2>Cuando dos DFT son dadas (secuencias de tamaño finito usualmente del tamaño <SPAN class=mrow id=id7013753><SPAN class=mrow id=id7013779><SPAN class=mi1>N</SPAN></SPAN></SPAN>), nosotros no podemos multiplicar esas dos señales así como así, como lo sugiere la formula de arriba usualmente conocida como <SPAN class=term>convolución linear</SPAN>. Ya que las DFT son periódicas, tienen valores no cero para <SPAN class=mrow id=id7013929><SPAN class=mrow id=id7013934><SPAN class=mrow id=id7013937><SPAN class=mi1>n</SPAN><SPAN class=mo id=id7013941>≥</SPAN><SPAN class=mi1>N</SPAN></SPAN></SPAN></SPAN> así la multiplicación de estas dos señales será no cero para <SPAN class=mrow id=id7013963><SPAN class=mrow id=id7013967><SPAN class=mrow id=id7013969><SPAN class=mi1>n</SPAN><SPAN class=mo id=id7013973>≥</SPAN><SPAN class=mi1>N</SPAN></SPAN></SPAN></SPAN>. Necesitamos definir otro tipo de convolucion que dará como resultado nuestra señal convuelta teniendo el valor de cero fuera del rango <SPAN class=mrow id=id7013996><SPAN class=mrow id=id7014000><SPAN class=mrow id=id7014006><SPAN class=mi1>n</SPAN><SPAN class=mo id=id7014010>=</SPAN><SPAN class=mrow id=id7014012><SPAN id=id7014013L>{</SPAN><SPAN id=id7014013M><SPAN class=mn>0</SPAN><SPAN id=id7014016X1>,</SPAN><SPAN class=mn>1</SPAN><SPAN id=id7014019X2>,</SPAN><SPAN class=mi1>…</SPAN><SPAN id=id7014021X3>,</SPAN><SPAN class=mrow id=id7014027><SPAN class=mi1>N</SPAN><SPAN class=mo id=id7014030>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SPAN><SPAN id=id7014013R>}</SPAN> <SCRIPT type=text/javascript> var mrowH = id7014013M.offsetHeight; mrowStretch(id7014013L,'ì','ï','í','î'); mrowStretch(id7014013R,'ü','ú','ý','þ');</SCRIPT> </SPAN></SPAN></SPAN></SPAN>. Esto nos ayuda a desarrollar la idea de <SPAN class=term>convolución circular</SPAN>, también conocida como convolución cíclica o periódica. </FONT></DIV> <DIV class=section id=sec2> <H2><FONT size=2>Formula de la Convolución Circular</FONT></H2> <DIV class=para id=para1><FONT size=2>¿Qué pasa cuando multiplicamos dos DFT una con la otra, donde <SPAN class=mrow id=id7014086><SPAN class=mrow id=id7014091><SPAN class=mrow id=id7014095><SPAN class=mi1>Y</SPAN><SPAN class=mo id=id7014098></SPAN><SPAN id=id7014100L>[</SPAN><SPAN id=id7014100M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7014100R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7014100M.offsetHeight; mrowStretch(id7014100L,'é','ê','ê','ë'); mrowStretch(id7014100R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>es la DFT de <SPAN class=mrow id=id7014127><SPAN class=mrow id=id7014129><SPAN class=mrow id=id7014134><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7014137></SPAN><SPAN id=id7014139L>[</SPAN><SPAN id=id7014139M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7014139R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7014139M.offsetHeight; mrowStretch(id7014139L,'é','ê','ê','ë'); mrowStretch(id7014139R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>? </FONT> <DIV class=equation id=eqa><FONT size=2><SPAN class=mrow id=id7014170><SPAN class=mrow id=id7014174><SPAN class=mrow id=id7014182><SPAN class=mrow id=id7014183><SPAN class=mi1>Y</SPAN><SPAN class=mo id=id7014186></SPAN><SPAN id=id7014188L>[</SPAN><SPAN id=id7014188M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7014188R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7014188M.offsetHeight; mrowStretch(id7014188L,'é','ê','ê','ë'); mrowStretch(id7014188R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN><SPAN class=mrow id=id7014195><SPAN class=mo id=id7014196>=</SPAN></SPAN><SPAN class=mrow id=id7014198><SPAN class=mrow id=id7014201><SPAN class=mrow id=id7014204><SPAN class=mi1>F</SPAN><SPAN class=mo id=id7014207></SPAN><SPAN id=id7014210L>[</SPAN><SPAN id=id7014210M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7014210R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7014210M.offsetHeight; mrowStretch(id7014210L,'é','ê','ê','ë'); mrowStretch(id7014210R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7014218></SPAN><SPAN class=mrow id=id7014220><SPAN class=mi1>H</SPAN><SPAN class=mo id=id7014223></SPAN><SPAN id=id7014226L>[</SPAN><SPAN id=id7014226M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7014226R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7014226M.offsetHeight; mrowStretch(id7014226L,'é','ê','ê','ë'); mrowStretch(id7014226R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN></SPAN></SPAN><SPAN class=equation-number>(2)</SPAN> </FONT></DIV><FONT size=2>cuando <SPAN class=mrow id=id7014177><SPAN class=mrow id=id7014181><SPAN class=mrow id=id7014307><SPAN class=mn>0</SPAN><SPAN class=mo id=id7014311>≤</SPAN><SPAN class=mi1>k</SPAN><SPAN class=mo id=id7014316>≤</SPAN><SPAN class=mrow id=id7014322><SPAN class=mi1>N</SPAN><SPAN class=mo id=id7014326>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SPAN></SPAN></SPAN> </FONT></DIV> <DIV class=para id=para2><FONT size=2>Usando la formula sintetizada de DFT para <SPAN class=mrow id=id7014370><SPAN class=mrow id=id7014376><SPAN class=mrow id=id7014380><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7014383></SPAN><SPAN id=id7014386L>[</SPAN><SPAN id=id7014386M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7014386R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7014386M.offsetHeight; mrowStretch(id7014386L,'é','ê','ê','ë'); mrowStretch(id7014386R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN></FONT> <DIV class=equation id=eq2><SPAN class=mrow id=id7014419><SPAN class=mrow id=id7014424><FONT size=2><SPAN class=mrow id=id7014438><SPAN class=mrow id=id7014439><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7014442></SPAN><SPAN id=id7014444L>[</SPAN><SPAN id=id7014444M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7014444R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7014444M.offsetHeight; mrowStretch(id7014444L,'é','ê','ê','ë'); mrowStretch(id7014444R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN><SPAN class=mrow id=id7014451><SPAN class=mo id=id7014452>=</SPAN></SPAN></FONT><SPAN class=mrow id=id7014454><SPAN class=mrow id=id7014458><SPAN class=mrow id=id7014460><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7014462> <TD class=mfracaa><SPAN class=mrow id=id7014463><SPAN class=mn><FONT size=2>1</FONT></SPAN></SPAN></TD></TR> <TR class=mfracb id=bid7014462 style="HEIGHT: 20px"> <TD><SPAN class=mrow id=id7014467><SPAN class=mi1><FONT size=2>N</FONT></SPAN></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7014462.offsetHeight > bid7014462.offsetHeight ) bid7014462.style.setExpression("height",aid7014462.offsetHeight ); else aid7014462.style.setExpression("height",bid7014462.offsetHeight ); </SCRIPT> </SPAN><SPAN class=mo id=id7014471></SPAN><SPAN class=mrow id=id7014473> <TABLE class=munderover> <TBODY> <TR> <TD><SPAN class=mrow id=id7014489><SPAN class=mrow id=id7014496><FONT size=2><SPAN class=mi1>N</SPAN><SPAN class=mo id=id7014500>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR> <TR> <TD><SPAN class=mo id=id7014476><FONT size=2>∑</FONT></SPAN></TD></TR> <TR> <TD><SPAN class=mrow id=id7014478><FONT size=2><SPAN class=mi1>k</SPAN><SPAN class=mo id=id7014484>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><SPAN id=id7014505L> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>æ</FONT><BR><FONT face=symbol>è</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id7014505M><SPAN class=mrow id=id7014509><FONT size=2><SPAN class=mrow id=id7014513><SPAN class=mi1>F</SPAN><SPAN class=mo id=id7014516></SPAN><SPAN id=id7014520L>[</SPAN><SPAN id=id7014520M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7014520R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7014520M.offsetHeight; mrowStretch(id7014520L,'é','ê','ê','ë'); mrowStretch(id7014520R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7014527></SPAN><SPAN class=mrow id=id7014529><SPAN class=mi1>H</SPAN><SPAN class=mo id=id7014532></SPAN><SPAN id=id7014535L>[</SPAN><SPAN id=id7014535M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7014535R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7014535M.offsetHeight; mrowStretch(id7014535L,'é','ê','ê','ë'); mrowStretch(id7014535R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7014542></SPAN><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN></FONT><SUP><SPAN class=mrow id=id7014553><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id7014556></SPAN></FONT><SPAN class=mrow id=id7014559><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7014560> <TD class=mfracaa><SPAN class=mrow id=id7014562><SPAN class=mrow id=id7014565><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7014572></SPAN><SPAN class=mi1>π</SPAN></FONT></SPAN></SPAN></TD></TR> <TR class=mfracb id=bid7014560 style="HEIGHT: 18px"> <TD><SPAN class=mrow id=id7014575><SPAN class=mi1><FONT size=2>N</FONT></SPAN></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7014560.offsetHeight > bid7014560.offsetHeight ) bid7014560.style.setExpression("height",aid7014560.offsetHeight ); else aid7014560.style.setExpression("height",bid7014560.offsetHeight ); </SCRIPT> </SPAN><SPAN class=mo id=id7014579></SPAN><FONT size=2><SPAN class=mi1>k</SPAN><SPAN class=mo id=id7014583></SPAN><SPAN class=mi1>n</SPAN></FONT></SPAN></SUP></SPAN></SPAN><SPAN id=id7014505R><FONT size=2> </FONT> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>ö</FONT><BR><FONT face=symbol>ø</FONT></FONT></TD></TR></TBODY></TABLE></SPAN> <SCRIPT type=text/javascript> var mrowH = id7014505M.offsetHeight; mrowStretch(id7014505L,'æ','ç','ç','è'); mrowStretch(id7014505R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN></SPAN></SPAN><FONT size=2><SPAN class=equation-number>(3)</SPAN> </FONT></DIV></DIV> <DIV class=para id=para3><FONT size=2>Y aplicando análisis a la formula </FONT><SPAN class=mrow id=id7014815><SPAN class=mrow id=id7014740><SPAN class=mrow id=id7014761><FONT size=2><SPAN class=mrow id=id7014762><SPAN class=mi1>F</SPAN><SPAN class=mo id=id7014765></SPAN><SPAN id=id7014767L>[</SPAN><SPAN id=id7014767M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7014767R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7014767M.offsetHeight; mrowStretch(id7014767L,'é','ê','ê','ë'); mrowStretch(id7014767R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7014774>=</SPAN></FONT><SPAN class=mrow id=id7014777><FONT size=2> </FONT> <TABLE class=munderover> <TBODY> <TR> <TD><SPAN class=mrow id=id7014792><SPAN class=mrow id=id7014797><FONT size=2><SPAN class=mi1>N</SPAN><SPAN class=mo id=id7014800>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR> <TR> <TD><SPAN class=mo id=id7014779><FONT size=2>∑</FONT></SPAN></TD></TR> <TR> <TD><SPAN class=mrow id=id7014781><FONT size=2><SPAN class=mi1>m</SPAN><SPAN class=mo id=id7014787>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><SPAN id=id7014819L> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>æ</FONT><BR><FONT face=symbol>è</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id7014819M><SPAN class=mrow id=id7014823><FONT size=2><SPAN class=mrow id=id7014827><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7014831></SPAN><SPAN id=id7014834L>[</SPAN><SPAN id=id7014834M><SPAN class=mi1>m</SPAN></SPAN><SPAN id=id7014834R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7014834M.offsetHeight; mrowStretch(id7014834L,'é','ê','ê','ë'); mrowStretch(id7014834R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7014841></SPAN><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN></FONT><SUP><SPAN class=mrow id=id7014852><FONT size=2><SPAN id=id7014853L>(</SPAN><SPAN id=id7014853M><SPAN class=mrow id=id7014859><SPAN class=mo id=id7014860>−</SPAN><SPAN class=mi1>j</SPAN></SPAN></SPAN><SPAN id=id7014853R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7014853M.offsetHeight; mrowStretch(id7014853L,'æ','ç','ç','è'); mrowStretch(id7014853R,'ö','÷','÷','ø');</SCRIPT> <SPAN class=mo id=id7014864></SPAN></FONT><SPAN class=mrow id=id7014867> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7014868> <TD class=mfracaa><SPAN class=mrow id=id7014869><SPAN class=mrow id=id7014872><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7014879></SPAN><SPAN class=mi1>π</SPAN></FONT></SPAN></SPAN></TD></TR> <TR class=mfracb id=bid7014868 style="HEIGHT: 18px"> <TD><SPAN class=mrow id=id7014882><SPAN class=mi1><FONT size=2>N</FONT></SPAN></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7014868.offsetHeight > bid7014868.offsetHeight ) bid7014868.style.setExpression("height",aid7014868.offsetHeight ); else aid7014868.style.setExpression("height",bid7014868.offsetHeight ); </SCRIPT> </SPAN><SPAN class=mo id=id7014886></SPAN><FONT size=2><SPAN class=mi1>k</SPAN><SPAN class=mo id=id7014889></SPAN><SPAN class=mi1>n</SPAN></FONT></SPAN></SUP></SPAN></SPAN><SPAN id=id7014819R><FONT size=2> </FONT> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>ö</FONT><BR><FONT face=symbol>ø</FONT></FONT></TD></TR></TBODY></TABLE></SPAN> <SCRIPT type=text/javascript> var mrowH = id7014819M.offsetHeight; mrowStretch(id7014819L,'æ','ç','ç','è'); mrowStretch(id7014819R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN></SPAN> <DIV class=equation id=eq3><SPAN class=mrow id=id7014795><SPAN class=mrow id=id7014875> <TABLE class=mtable> <TBODY> <TR> <TD><SPAN class=mrow id=id7015056><SPAN class=mrow id=id7015057><FONT size=2><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7015060></SPAN><SPAN id=id7015063L>[</SPAN><SPAN id=id7015063M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7015063R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7015063M.offsetHeight; mrowStretch(id7015063L,'é','ê','ê','ë'); mrowStretch(id7015063R,'ù','ú','ú','û');</SCRIPT> </FONT></SPAN></SPAN></TD> <TD><SPAN class=mo id=id7015073><FONT size=2>=</FONT></SPAN></TD> <TD><SPAN class=mrow id=id7015079><SPAN class=mrow id=id7015082><SPAN class=mrow id=id7015087> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7015088 style="HEIGHT: 0px"> <TD class=mfracaa><SPAN class=mrow id=id7015089><SPAN class=mn><FONT size=2>1</FONT></SPAN></SPAN></TD></TR> <TR class=mfracb id=bid7015088> <TD><SPAN class=mrow id=id7015093><SPAN class=mi1><FONT size=2>N</FONT></SPAN></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7015088.offsetHeight > bid7015088.offsetHeight ) bid7015088.style.setExpression("height",aid7015088.offsetHeight ); else aid7015088.style.setExpression("height",bid7015088.offsetHeight ); </SCRIPT> </SPAN><SPAN class=mo id=id7015097></SPAN><SPAN class=mrow id=id7015099> <TABLE class=munderover> <TBODY> <TR> <TD><SPAN class=mrow id=id7015115><SPAN class=mrow id=id7015122><FONT size=2><SPAN class=mi1>N</SPAN><SPAN class=mo id=id7015125>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR> <TR> <TD><SPAN class=mo id=id7015102><FONT size=2>∑</FONT></SPAN></TD></TR> <TR> <TD><SPAN class=mrow id=id7015104><FONT size=2><SPAN class=mi1>k</SPAN><SPAN class=mo id=id7015110>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><SPAN id=id7015131L><FONT size=2>(</FONT></SPAN><SPAN id=id7015131M><SPAN class=mrow id=id7015135><SPAN id=id7015136L><FONT size=2>(</FONT></SPAN><SPAN id=id7015136M><SPAN class=mrow id=id7015144><FONT size=2> </FONT> <TABLE class=munderover> <TBODY> <TR> <TD><SPAN class=mrow id=id7015157><SPAN class=mrow id=id7015164><FONT size=2><SPAN class=mi1>N</SPAN><SPAN class=mo id=id7015167>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR> <TR> <TD><SPAN class=mo id=id7015146><FONT size=2>∑</FONT></SPAN></TD></TR> <TR> <TD><SPAN class=mrow id=id7015148><FONT size=2><SPAN class=mi1>m</SPAN><SPAN class=mo id=id7015152>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><SPAN id=id7015173L><FONT size=2>(</FONT></SPAN><SPAN id=id7015173M><SPAN class=mrow id=id7015177><FONT size=2><SPAN class=mrow id=id7015181><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7015184></SPAN><SPAN id=id7015187L>[</SPAN><SPAN id=id7015187M><SPAN class=mi1>m</SPAN></SPAN><SPAN id=id7015187R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7015187M.offsetHeight; mrowStretch(id7015187L,'é','ê','ê','ë'); mrowStretch(id7015187R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7015195></SPAN><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN></FONT><SUP><SPAN class=mrow id=id7015205><FONT size=2><SPAN id=id7015206L>(</SPAN><SPAN id=id7015206M><SPAN class=mrow id=id7015212><SPAN class=mo id=id7015213>−</SPAN><SPAN class=mi1>j</SPAN></SPAN></SPAN><SPAN id=id7015206R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7015206M.offsetHeight; mrowStretch(id7015206L,'æ','ç','ç','è'); mrowStretch(id7015206R,'ö','÷','÷','ø');</SCRIPT> <SPAN class=mo id=id7015221></SPAN></FONT><SPAN class=mrow id=id7015223> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7015224 style="HEIGHT: 0px"> <TD class=mfracaa><SPAN class=mrow id=id7015226><SPAN class=mrow id=id7015228><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7015235></SPAN><SPAN class=mi1>π</SPAN></FONT></SPAN></SPAN></TD></TR> <TR class=mfracb id=bid7015224> <TD><SPAN class=mrow id=id7015239><SPAN class=mi1><FONT size=2>N</FONT></SPAN></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7015224.offsetHeight > bid7015224.offsetHeight ) bid7015224.style.setExpression("height",aid7015224.offsetHeight ); else aid7015224.style.setExpression("height",bid7015224.offsetHeight ); </SCRIPT> </SPAN><SPAN class=mo id=id7015242></SPAN><FONT size=2><SPAN class=mi1>k</SPAN><SPAN class=mo id=id7015246></SPAN><SPAN class=mi1>n</SPAN></FONT></SPAN></SUP></SPAN></SPAN><FONT size=2><SPAN id=id7015173R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7015173M.offsetHeight; mrowStretch(id7015173L,'æ','ç','ç','è'); mrowStretch(id7015173R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN><FONT size=2><SPAN id=id7015136R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7015136M.offsetHeight; mrowStretch(id7015136L,'æ','ç','ç','è'); mrowStretch(id7015136R,'ö','÷','÷','ø');</SCRIPT> <SPAN class=mo id=id7015250></SPAN><SPAN class=mrow id=id7015251><SPAN class=mi1>H</SPAN><SPAN class=mo id=id7015254></SPAN><SPAN id=id7015257L>[</SPAN><SPAN id=id7015257M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7015257R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7015257M.offsetHeight; mrowStretch(id7015257L,'é','ê','ê','ë'); mrowStretch(id7015257R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7015264></SPAN><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN></FONT><SUP><SPAN class=mrow id=id7015271><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id7015274></SPAN></FONT><SPAN class=mrow id=id7015276><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7015278 style="HEIGHT: 0px"> <TD class=mfracaa><SPAN class=mrow id=id7015279><SPAN class=mrow id=id7015281><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7015284></SPAN><SPAN class=mi1>π</SPAN></FONT></SPAN></SPAN></TD></TR> <TR class=mfracb id=bid7015278> <TD><SPAN class=mrow id=id7015288><SPAN class=mi1><FONT size=2>N</FONT></SPAN></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7015278.offsetHeight > bid7015278.offsetHeight ) bid7015278.style.setExpression("height",aid7015278.offsetHeight ); else aid7015278.style.setExpression("height",bid7015278.offsetHeight ); </SCRIPT> </SPAN><SPAN class=mo id=id7015291></SPAN><FONT size=2><SPAN class=mi1>k</SPAN><SPAN class=mo id=id7015295></SPAN><SPAN class=mi1>n</SPAN></FONT></SPAN></SUP></SPAN></SPAN><FONT size=2><SPAN id=id7015131R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7015131M.offsetHeight; mrowStretch(id7015131L,'æ','ç','ç','è'); mrowStretch(id7015131R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN></TD></TR> <TR> <TD><FONT size=2></FONT></TD> <TD><SPAN class=mo id=id7015306><FONT size=2>=</FONT></SPAN></TD> <TD><SPAN class=mrow id=id7015312><SPAN class=mrow id=id7015315> <TABLE class=munderover> <TBODY> <TR> <TD><SPAN class=mrow id=id7015329><SPAN class=mrow id=id7015332><FONT size=2><SPAN class=mi1>N</SPAN><SPAN class=mo id=id7015335>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR> <TR> <TD><SPAN class=mo id=id7015317><FONT size=2>∑</FONT></SPAN></TD></TR> <TR> <TD><SPAN class=mrow id=id7015319><FONT size=2><SPAN class=mi1>m</SPAN><SPAN class=mo id=id7015324>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><SPAN id=id7015341L><FONT size=2>(</FONT></SPAN><SPAN id=id7015341M><SPAN class=mrow id=id7015345><FONT size=2><SPAN class=mrow id=id7015346><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7015350></SPAN><SPAN id=id7015352L>[</SPAN><SPAN id=id7015352M><SPAN class=mi1>m</SPAN></SPAN><SPAN id=id7015352R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7015352M.offsetHeight; mrowStretch(id7015352L,'é','ê','ê','ë'); mrowStretch(id7015352R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7015359></SPAN><SPAN id=id7015361L>(</SPAN></FONT><SPAN id=id7015361M><SPAN class=mrow id=id7015365><SPAN class=mrow id=id7015367><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7015368 style="HEIGHT: 0px"> <TD class=mfracaa><SPAN class=mrow id=id7015369><SPAN class=mn><FONT size=2>1</FONT></SPAN></SPAN></TD></TR> <TR class=mfracb id=bid7015368> <TD><SPAN class=mrow id=id7015373><SPAN class=mi1><FONT size=2>N</FONT></SPAN></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7015368.offsetHeight > bid7015368.offsetHeight ) bid7015368.style.setExpression("height",aid7015368.offsetHeight ); else aid7015368.style.setExpression("height",bid7015368.offsetHeight ); </SCRIPT> </SPAN><SPAN class=mo id=id7015377></SPAN><SPAN class=mrow id=id7015379> <TABLE class=munderover> <TBODY> <TR> <TD><SPAN class=mrow id=id7015393><SPAN class=mrow id=id7015396><FONT size=2><SPAN class=mi1>N</SPAN><SPAN class=mo id=id7015399>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR> <TR> <TD><SPAN class=mo id=id7015381><FONT size=2>∑</FONT></SPAN></TD></TR> <TR> <TD><SPAN class=mrow id=id7015383><FONT size=2><SPAN class=mi1>k</SPAN><SPAN class=mo id=id7015387>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><SPAN id=id7015404L><FONT size=2>(</FONT></SPAN><SPAN id=id7015404M><SPAN class=mrow id=id7015408><FONT size=2><SPAN class=mrow id=id7015410><SPAN class=mi1>H</SPAN><SPAN class=mo id=id7015413></SPAN><SPAN id=id7015416L>[</SPAN><SPAN id=id7015416M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7015416R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7015416M.offsetHeight; mrowStretch(id7015416L,'é','ê','ê','ë'); mrowStretch(id7015416R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7015423></SPAN><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN></FONT><SUP><SPAN class=mrow id=id7015430><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id7015433></SPAN></FONT><SPAN class=mrow id=id7015436><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7015437 style="HEIGHT: 0px"> <TD class=mfracaa><SPAN class=mrow id=id7015438><SPAN class=mrow id=id7015440><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7015445></SPAN><SPAN class=mi1>π</SPAN></FONT></SPAN></SPAN></TD></TR> <TR class=mfracb id=bid7015437> <TD><SPAN class=mrow id=id7015448><SPAN class=mi1><FONT size=2>N</FONT></SPAN></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7015437.offsetHeight > bid7015437.offsetHeight ) bid7015437.style.setExpression("height",aid7015437.offsetHeight ); else aid7015437.style.setExpression("height",bid7015437.offsetHeight ); </SCRIPT> </SPAN><SPAN class=mo id=id7015452></SPAN><FONT size=2><SPAN class=mi1>k</SPAN><SPAN class=mo id=id7015456></SPAN><SPAN id=id7015457L>(</SPAN><SPAN id=id7015457M><SPAN class=mrow id=id7015462><SPAN class=mi1>n</SPAN><SPAN class=mo id=id7015465>−</SPAN><SPAN class=mi1>m</SPAN></SPAN></SPAN><SPAN id=id7015457R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7015457M.offsetHeight; mrowStretch(id7015457L,'æ','ç','ç','è'); mrowStretch(id7015457R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SUP></SPAN></SPAN><FONT size=2><SPAN id=id7015404R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7015404M.offsetHeight; mrowStretch(id7015404L,'æ','ç','ç','è'); mrowStretch(id7015404R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN><FONT size=2><SPAN id=id7015361R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7015361M.offsetHeight; mrowStretch(id7015361L,'æ','ç','ç','è'); mrowStretch(id7015361R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN><FONT size=2><SPAN id=id7015341R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7015341M.offsetHeight; mrowStretch(id7015341L,'æ','ç','ç','è'); mrowStretch(id7015341R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript></SCRIPT> </SPAN></SPAN><FONT size=2><SPAN class=equation-number>(4)</SPAN> </FONT></DIV><FONT size=2>donde podemos reducir la segunda sumatoria de la ecuación de arriba en </FONT><SPAN class=mrow id=id7014746><SPAN class=mrow id=id7015200><SPAN class=mrow id=id7015210><FONT size=2><SPAN class=mrow id=id7015046><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7015199></SPAN><SPAN id=id7015118L>[</SPAN><SPAN id=id7015118M><SPAN class=mrow id=id7015978><SPAN><SPAN style="COLOR: red"><apply></SPAN><SPAN style="COLOR: red"><ci></SPAN><SPAN class=mo id=id7015986>(</SPAN><SPAN class=mo id=id7015990>(</SPAN><SPAN class=mi1>n</SPAN><SPAN class=mo id=id7015998>−</SPAN><SPAN class=mi1>m</SPAN><SPAN class=mo id=id7016006>)</SPAN><SPAN class=mo id=id7016010>)</SPAN><SPAN style="COLOR: red"></ci></SPAN><SPAN style="COLOR: red"></apply></SPAN></SPAN><SUB><SPAN class=mi1>N</SPAN></SUB></SPAN></SPAN><SPAN id=id7015118R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7015118M.offsetHeight; mrowStretch(id7015118L,'é','ê','ê','ë'); mrowStretch(id7015118R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7016021>=</SPAN></FONT><SPAN class=mrow id=id7016024><SPAN class=mrow id=id7016028><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7016029> <TD class=mfracaa><SPAN class=mrow id=id7016030><SPAN class=mn><FONT size=2>1</FONT></SPAN></SPAN></TD></TR> <TR class=mfracb id=bid7016029 style="HEIGHT: 20px"> <TD><SPAN class=mrow id=id7016034><SPAN class=mi1><FONT size=2>N</FONT></SPAN></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7016029.offsetHeight > bid7016029.offsetHeight ) bid7016029.style.setExpression("height",aid7016029.offsetHeight ); else aid7016029.style.setExpression("height",bid7016029.offsetHeight ); </SCRIPT> </SPAN><SPAN class=mo id=id7016038></SPAN><SPAN class=mrow id=id7016041> <TABLE class=munderover> <TBODY> <TR> <TD><SPAN class=mrow id=id7016057><SPAN class=mrow id=id7016064><FONT size=2><SPAN class=mi1>N</SPAN><SPAN class=mo id=id7016067>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR> <TR> <TD><SPAN class=mo id=id7016043><FONT size=2>∑</FONT></SPAN></TD></TR> <TR> <TD><SPAN class=mrow id=id7016046><FONT size=2><SPAN class=mi1>k</SPAN><SPAN class=mo id=id7016051>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><SPAN id=id7016072L> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>æ</FONT><BR><FONT face=symbol>è</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id7016072M><SPAN class=mrow id=id7016076><FONT size=2><SPAN class=mrow id=id7016080><SPAN class=mi1>H</SPAN><SPAN class=mo id=id7016084></SPAN><SPAN id=id7016087L>[</SPAN><SPAN id=id7016087M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7016087R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7016087M.offsetHeight; mrowStretch(id7016087L,'é','ê','ê','ë'); mrowStretch(id7016087R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7016094></SPAN><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN></FONT><SUP><SPAN class=mrow id=id7016105><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id7016108></SPAN></FONT><SPAN class=mrow id=id7016111><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7016112> <TD class=mfracaa><SPAN class=mrow id=id7016114><SPAN class=mrow id=id7016117><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7016124></SPAN><SPAN class=mi1>π</SPAN></FONT></SPAN></SPAN></TD></TR> <TR class=mfracb id=bid7016112 style="HEIGHT: 18px"> <TD><SPAN class=mrow id=id7016127><SPAN class=mi1><FONT size=2>N</FONT></SPAN></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7016112.offsetHeight > bid7016112.offsetHeight ) bid7016112.style.setExpression("height",aid7016112.offsetHeight ); else aid7016112.style.setExpression("height",bid7016112.offsetHeight ); </SCRIPT> </SPAN><SPAN class=mo id=id7016131></SPAN><FONT size=2><SPAN class=mi1>k</SPAN><SPAN class=mo id=id7016135></SPAN><SPAN id=id7016136L>(</SPAN><SPAN id=id7016136M><SPAN class=mrow id=id7016141><SPAN class=mi1>n</SPAN><SPAN class=mo id=id7016144>−</SPAN><SPAN class=mi1>m</SPAN></SPAN></SPAN><SPAN id=id7016136R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7016136M.offsetHeight; mrowStretch(id7016136L,'æ','ç','ç','è'); mrowStretch(id7016136R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SUP></SPAN></SPAN><SPAN id=id7016072R> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>ö</FONT><BR><FONT face=symbol>ø</FONT></FONT></TD></TR></TBODY></TABLE></SPAN> <SCRIPT type=text/javascript> var mrowH = id7016072M.offsetHeight; mrowStretch(id7016072L,'æ','ç','ç','è'); mrowStretch(id7016072R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN></SPAN></SPAN><SPAN class=mrow id=id7016367><SPAN class=mrow id=id7016026><SPAN class=mrow id=id7016115><FONT size=2><SPAN class=mrow id=id7015085><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7016120></SPAN><SPAN id=id7016059L>[</SPAN><SPAN id=id7016059M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7016059R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7016059M.offsetHeight; mrowStretch(id7016059L,'é','ê','ê','ë'); mrowStretch(id7016059R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7016373>=</SPAN></FONT><SPAN class=mrow id=id7016375><FONT size=2> </FONT> <TABLE class=munderover> <TBODY> <TR> <TD><SPAN class=mrow id=id7016389><SPAN class=mrow id=id7016394><FONT size=2><SPAN class=mi1>N</SPAN><SPAN class=mo id=id7016397>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR> <TR> <TD><SPAN class=mo id=id7016378><FONT size=2>∑</FONT></SPAN></TD></TR> <TR> <TD><SPAN class=mrow id=id7016380><FONT size=2><SPAN class=mi1>m</SPAN><SPAN class=mo id=id7016384>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><FONT size=2><SPAN id=id7016402L>(</SPAN><SPAN id=id7016402M><SPAN class=mrow id=id7016406><SPAN class=mrow id=id7016410><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7016414></SPAN><SPAN id=id7016417L>[</SPAN><SPAN id=id7016417M><SPAN class=mi1>m</SPAN></SPAN><SPAN id=id7016417R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7016417M.offsetHeight; mrowStretch(id7016417L,'é','ê','ê','ë'); mrowStretch(id7016417R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7016424></SPAN><SPAN class=mrow id=id7016426><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7016430></SPAN><SPAN id=id7016432L>[</SPAN><SPAN id=id7016432M><SPAN class=mrow id=id7016437><SPAN><SPAN style="COLOR: red"><apply></SPAN><SPAN style="COLOR: red"><ci></SPAN><SPAN class=mo id=id7016446>(</SPAN><SPAN class=mo id=id7016450>(</SPAN><SPAN class=mi1>n</SPAN><SPAN class=mo id=id7016458>−</SPAN><SPAN class=mi1>m</SPAN><SPAN class=mo id=id7016466>)</SPAN><SPAN class=mo id=id7016470>)</SPAN><SPAN style="COLOR: red"></ci></SPAN><SPAN style="COLOR: red"></apply></SPAN></SPAN><SUB><SPAN class=mi1>N</SPAN></SUB></SPAN></SPAN><SPAN id=id7016432R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7016432M.offsetHeight; mrowStretch(id7016432L,'é','ê','ê','ë'); mrowStretch(id7016432R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN><SPAN id=id7016402R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7016402M.offsetHeight; mrowStretch(id7016402L,'æ','ç','ç','è'); mrowStretch(id7016402R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN></SPAN><FONT size=2>Igual a la convolución circular! cuando tenemos <SPAN class=mrow id=id7016633><SPAN class=mrow id=id7016392><SPAN class=mrow id=id7016634><SPAN class=mn>0</SPAN><SPAN class=mo id=id7016638>≤</SPAN><SPAN class=mi1>n</SPAN><SPAN class=mo id=id7016643>≤</SPAN><SPAN class=mrow id=id7016646><SPAN class=mi1>N</SPAN><SPAN class=mo id=id7016650>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SPAN></SPAN></SPAN> arriba , para obtener dos: </FONT> <DIV class=equation id=eq4><FONT size=2><SPAN class=mrow id=id7016693><SPAN class=mrow id=id7016700><SPAN class=mrow id=id7016705><SPAN class=mrow id=id7016711><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7016714></SPAN><SPAN id=id7016718L>[</SPAN><SPAN id=id7016718M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7016718R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7016718M.offsetHeight; mrowStretch(id7016718L,'é','ê','ê','ë'); mrowStretch(id7016718R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7016724>≡</SPAN><SPAN class=mrow id=id7016727><SPAN id=id7016728L>(</SPAN><SPAN id=id7016728M><SPAN class=mrow id=id7016739><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7016742></SPAN><SPAN id=id7016745L>[</SPAN><SPAN id=id7016745M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7016745R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7016745M.offsetHeight; mrowStretch(id7016745L,'é','ê','ê','ë'); mrowStretch(id7016745R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN id=id7016739X1>,</SPAN><SPAN class=mo id=id7016753>⊛</SPAN><SPAN id=id7016753X2>,</SPAN><SPAN class=mrow id=id7016756><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7016759></SPAN><SPAN id=id7016762L>[</SPAN><SPAN id=id7016762M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7016762R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7016762M.offsetHeight; mrowStretch(id7016762L,'é','ê','ê','ë'); mrowStretch(id7016762R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN><SPAN id=id7016728R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7016728M.offsetHeight; mrowStretch(id7016728L,'æ','ç','ç','è'); mrowStretch(id7016728R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN></SPAN><SPAN class=equation-number>(5)</SPAN> </FONT></DIV> <DIV class=note><FONT size=2><SPAN class=note-before>note: </SPAN>Que la notación <SPAN class=mrow id=id7016848><SPAN class=mrow id=id7016855><SPAN class=mo id=id7016857>⊛</SPAN></SPAN></SPAN> representa la convolucion circular "mod N". </FONT></DIV></DIV> <DIV class=section id=sub1> <H2><FONT size=2>Pasos para la Convolución Circular</FONT></H2> <DIV class=para id=para4><FONT size=2>Los pasos a seguir para la convolucion cíclica son los mismos que se usan en la convolución linear, excepto que todos los cálculos para todos los índices están hecho"mod N" = "en la rueda" </FONT></DIV> <UL class=list id=step1><FONT size=2><SPAN class=name>Pasos para la Convolución Cíclica</SPAN> </FONT> <LI class=item><FONT size=2>Paso 1: "Grafique" <SPAN class=mrow id=id7016908><SPAN class=mrow id=id7016909><SPAN class=mrow id=id7016918><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7016922></SPAN><SPAN id=id7016925L>[</SPAN><SPAN id=id7016925M><SPAN class=mi1>m</SPAN></SPAN><SPAN id=id7016925R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7016925M.offsetHeight; mrowStretch(id7016925L,'é','ê','ê','ë'); mrowStretch(id7016925R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>y <SPAN class=mrow id=id7016953><SPAN class=mrow id=id7016915><SPAN class=mrow id=id7016963><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7016966></SPAN><SPAN id=id7016969L>[</SPAN><SPAN id=id7016969M><SPAN class=mrow id=id7016976><SPAN class=mrow id=id7016977><SPAN><SPAN style="COLOR: red"><apply></SPAN><SPAN class=mo id=id7016983>(</SPAN><SPAN class=mo id=id7016987>(</SPAN><SPAN class=mo id=id7016991>−</SPAN><SPAN class=mn>m</SPAN><SPAN class=mo id=id7016999>)</SPAN><SPAN class=mo id=id7017003>)</SPAN><SPAN style="COLOR: red"></apply></SPAN></SPAN><SUB><SPAN class=mi1>N</SPAN></SUB></SPAN></SPAN></SPAN><SPAN id=id7016969R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7016969M.offsetHeight; mrowStretch(id7016969L,'é','ê','ê','ë'); mrowStretch(id7016969R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN></FONT></LI></UL> <TABLE class=figure id=fig1 cellSpacing=0 cellPadding=0 width="50%" align=center border=0> <TBODY> <TR> <TD> <DIV class=inner-figure> <TABLE cellSpacing=5 cellPadding=0 width="100%" border=0> <TBODY> <TR> <TD vAlign=bottom><FONT size=2></FONT></TD> <TD vAlign=bottom><FONT size=2></FONT></TD></TR> <TR> <TD class=subfigure id=sub1_f1 vAlign=center><FONT size=2><IMG class=media src= http://www.wikilearning.com/imagescc/20470/cconv_s1.png∞ ></FONT></TD> <TD class=subfigure id=sub2_f1 vAlign=center><FONT size=2><IMG class=media src= http://www.wikilearning.com/imagescc/20470/cconv_s2.png∞ ></FONT></TD></TR> <TR> <TD vAlign=top> <DIV class=caption><SPAN class=caption-before><FONT size=2>Subfigure 1.1</FONT></SPAN></DIV></TD> <TD vAlign=top> <DIV class=caption><SPAN class=caption-before><FONT size=2>Subfigure 1.2</FONT></SPAN></DIV></TD></TR></TBODY></TABLE></DIV> <DIV class=caption><FONT size=2><SPAN class=caption-before>Figure 1: </SPAN>Step 1 </FONT></DIV></TD></TR></TBODY></TABLE> <UL class=list id=step2> <LI class=item><FONT size=2>Paso 2: "Rote" <SPAN class=mrow id=id7017169><SPAN class=mrow id=id7017173><SPAN class=mrow id=id7017182><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7017186></SPAN><SPAN id=id7017189L>[</SPAN><SPAN id=id7017189M><SPAN class=mrow id=id7017197><SPAN class=mrow id=id7017198><SPAN><SPAN style="COLOR: red"><apply></SPAN><SPAN class=mo id=id7017204>(</SPAN><SPAN class=mo id=id7017208>(</SPAN><SPAN class=mo id=id7017212>−</SPAN><SPAN class=mn>m</SPAN><SPAN class=mo id=id7017220>)</SPAN><SPAN class=mo id=id7017224>)</SPAN><SPAN style="COLOR: red"></apply></SPAN></SPAN><SUB><SPAN class=mi1>N</SPAN></SUB></SPAN></SPAN></SPAN><SPAN id=id7017189R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7017189M.offsetHeight; mrowStretch(id7017189L,'é','ê','ê','ë'); mrowStretch(id7017189R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN><SPAN class=mrow id=id7012544><SPAN class=mrow id=id7017178><SPAN class=mi1>n</SPAN></SPAN></SPAN> en la dirección ACW ( dirección opuesta al reloj) para obtener <SPAN class=mrow id=id7012561><SPAN class=mrow id=id7012565><SPAN class=mrow id=id7012571><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7012574></SPAN><SPAN id=id7012578L>[</SPAN><SPAN id=id7012578M><SPAN class=mrow id=id7012583><SPAN><SPAN style="COLOR: red"><apply></SPAN><SPAN class=mo id=id7012589>(</SPAN><SPAN class=mo id=id7012593>(</SPAN><SPAN class=mn>n</SPAN><SPAN class=mo id=id7012601>−</SPAN><SPAN class=mn>m</SPAN><SPAN class=mo id=id7012609>)</SPAN><SPAN class=mo id=id7012613>)</SPAN><SPAN style="COLOR: red"></apply></SPAN></SPAN><SUB><SPAN class=mi1>N</SPAN></SUB></SPAN></SPAN><SPAN id=id7012578R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7012578M.offsetHeight; mrowStretch(id7012578L,'é','ê','ê','ë'); mrowStretch(id7012578R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>(por ejemplo rote la secuencia, <SPAN class=mrow id=id7012683><SPAN class=mrow id=id7012686><SPAN class=mrow id=id7012692><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7012695></SPAN><SPAN id=id7012698L>[</SPAN><SPAN id=id7012698M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7012698R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7012698M.offsetHeight; mrowStretch(id7012698L,'é','ê','ê','ë'); mrowStretch(id7012698R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>, en dirección del reloj por <SPAN class=mrow id=id7012727><SPAN class=mrow id=id7012730><SPAN class=mi1>n</SPAN></SPAN></SPAN> pasos). </FONT></LI></UL> <TABLE class=figure id=fig2 cellSpacing=0 cellPadding=0 width="50%" align=center border=0> <TBODY> <TR> <TD> <DIV class=inner-figure><FONT size=2><IMG class=media src= http://www.wikilearning.com/imagescc/20470/cconv_s3.png∞ ></FONT></DIV> <DIV class=caption><FONT size=2><SPAN class=caption-before>Figure 2: </SPAN>Step 2 </FONT></DIV></TD></TR></TBODY></TABLE> <UL class=list id=step3> <LI class=item><FONT size=2>Paso 3: Multiplique punto por punto la rueda <SPAN class=mrow id=id7012791><SPAN class=mrow id=id7012794><SPAN class=mrow id=id7012803><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7012806></SPAN><SPAN id=id7012810L>[</SPAN><SPAN id=id7012810M><SPAN class=mi1>m</SPAN></SPAN><SPAN id=id7012810R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7012810M.offsetHeight; mrowStretch(id7012810L,'é','ê','ê','ë'); mrowStretch(id7012810R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>y la rueda <SPAN class=mrow id=id7012838><SPAN class=mrow id=id7012799><SPAN class=mrow id=id7012848><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7017656></SPAN><SPAN id=id7017658L>[</SPAN><SPAN id=id7017658M><SPAN class=mrow id=id7017662><SPAN><SPAN style="COLOR: red"><apply></SPAN><SPAN class=mo id=id7017669>(</SPAN><SPAN class=mo id=id7017673>(</SPAN><SPAN class=mn>n</SPAN><SPAN class=mo id=id7017681>−</SPAN><SPAN class=mn>m</SPAN><SPAN class=mo id=id7017689>)</SPAN><SPAN class=mo id=id7017693>)</SPAN><SPAN style="COLOR: red"></apply></SPAN></SPAN><SUB><SPAN class=mi1>N</SPAN></SUB></SPAN></SPAN><SPAN id=id7017658R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7017658M.offsetHeight; mrowStretch(id7017658L,'é','ê','ê','ë'); mrowStretch(id7017658R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>wheel. <SPAN class=mrow id=id7017762><SPAN class=mrow id=id7017764><SPAN class=mrow id=id7017777><SPAN class=mi>sum</SPAN><SPAN class=mo id=id7017780>=</SPAN><SPAN class=mrow id=id7017783><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7017786></SPAN><SPAN id=id7017789L>[</SPAN><SPAN id=id7017789M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7017789R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7017789M.offsetHeight; mrowStretch(id7017789L,'é','ê','ê','ë'); mrowStretch(id7017789R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN></SPAN></FONT></LI></UL> <UL class=list id=step4> <LI class=item><FONT size=2>Paso 4: Repite para <SPAN class=mrow id=id7017832><SPAN class=mrow id=id7017838><SPAN class=mrow id=id7017844><SPAN class=mn>0</SPAN><SPAN class=mo id=id7017848>≤</SPAN><SPAN class=mi1>n</SPAN><SPAN class=mo id=id7017853>≤</SPAN><SPAN class=mrow id=id7017859><SPAN class=mi1>N</SPAN><SPAN class=mo id=id7017863>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SPAN></SPAN></SPAN> </FONT></LI></UL> <DIV class=example id=mainExmpl><FONT size=2><SPAN class=example-before>Example 1: </SPAN><SPAN class=example-name>Convolve (n = 4) </SPAN> </FONT> <TABLE class=figure id=fig3 cellSpacing=0 cellPadding=0 width="50%" align=center border=0> <TBODY> <TR> <TD> <DIV class=inner-figure> <TABLE cellSpacing=5 cellPadding=0 width="100%" border=0> <TBODY> <TR> <TD vAlign=bottom><FONT size=2></FONT></TD> <TD vAlign=bottom><FONT size=2></FONT></TD></TR> <TR> <TD class=subfigure id=sub1_f3 vAlign=center><FONT size=2><IMG class=media src= http://www.wikilearning.com/imagescc/20470/cconv_p1.png∞ ></FONT></TD> <TD class=subfigure id=sub2_f3 vAlign=center><FONT size=2><IMG class=media src= http://www.wikilearning.com/imagescc/20470/cconv_p2.png∞ ></FONT></TD></TR> <TR> <TD vAlign=top> <DIV class=caption><SPAN class=caption-before><FONT size=2>Subfigure 3.1</FONT></SPAN></DIV></TD> <TD vAlign=top> <DIV class=caption><SPAN class=caption-before><FONT size=2>Subfigure 3.2</FONT></SPAN></DIV></TD></TR></TBODY></TABLE></DIV> <DIV class=caption><FONT size=2><SPAN class=caption-before>Figure 3: </SPAN>Dos señales discretas que seran convolucionadas. </FONT></DIV></TD></TR></TBODY></TABLE> <DIV class=para id=paraex1> <UL class=list id=eg_l1> <LI class=item><SPAN class=mrow id=id7018057><SPAN class=mrow id=id7018061><SPAN class=mrow id=id7018072><FONT size=2><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7018075></SPAN><SPAN id=id7018083L>[</SPAN><SPAN id=id7018083M><SPAN class=mrow id=id7018086><SPAN><SPAN style="COLOR: red"><apply></SPAN><SPAN class=mo id=id7018092>(</SPAN><SPAN class=mo id=id7018096>(</SPAN><SPAN class=mo id=id7018100>−</SPAN><SPAN class=mi1>m</SPAN><SPAN class=mo id=id7018108>)</SPAN><SPAN class=mo id=id7018112>)</SPAN><SPAN style="COLOR: red"></apply></SPAN></SPAN><SUB><SPAN class=mi1>N</SPAN></SUB></SPAN></SPAN><SPAN id=id7018083R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7018083M.offsetHeight; mrowStretch(id7018083L,'é','ê','ê','ë'); mrowStretch(id7018083R,'ù','ú','ú','û');</SCRIPT> </FONT></SPAN></SPAN></SPAN></LI></UL></DIV> <TABLE class=figure id=fig4 cellSpacing=0 cellPadding=0 width="50%" align=center border=0> <TBODY> <TR> <TD> <DIV class=inner-figure><FONT size=2><IMG class=media src= http://www.wikilearning.com/imagescc/20470/cconv_p3.png∞ ></FONT></DIV> <DIV class=caption><SPAN class=caption-before><FONT size=2>Figure 4</FONT></SPAN></DIV></TD></TR></TBODY></TABLE> <DIV class=para id=paraex2><FONT size=2>Multiplique <SPAN class=mrow id=id7018221><SPAN class=mrow id=id7018222><SPAN class=mrow id=id7018232><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7018235></SPAN><SPAN id=id7018238L>[</SPAN><SPAN id=id7018238M><SPAN class=mi1>m</SPAN></SPAN><SPAN id=id7018238R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7018238M.offsetHeight; mrowStretch(id7018238L,'é','ê','ê','ë'); mrowStretch(id7018238R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>y <SPAN class=mrow id=id7018266><SPAN class=mrow id=id7018267><SPAN class=mi>sume</SPAN></SPAN></SPAN> para dar: <SPAN class=mrow id=id7018288><SPAN class=mrow id=id7018292><SPAN class=mrow id=id7018303><SPAN class=mrow id=id7018304><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7018308></SPAN><SPAN id=id7018310L>[</SPAN><SPAN id=id7018310M><SPAN class=mn>0</SPAN></SPAN><SPAN id=id7018310R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7018310M.offsetHeight; mrowStretch(id7018310L,'é','ê','ê','ë'); mrowStretch(id7018310R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7018317>=</SPAN><SPAN class=mn>3</SPAN></SPAN></SPAN></SPAN> </FONT></DIV> <DIV class=para id=paraex3> <UL class=list id=eg_l2> <LI class=item><SPAN class=mrow id=id7018364><SPAN class=mrow id=id7018368><SPAN class=mrow id=id7018379><FONT size=2><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7018382></SPAN><SPAN id=id7018386L>[</SPAN><SPAN id=id7018386M><SPAN class=mrow id=id7018391><SPAN><SPAN style="COLOR: red"><apply></SPAN><SPAN class=mo id=id7018397>(</SPAN><SPAN class=mo id=id7018401>(</SPAN><SPAN class=mn>1</SPAN><SPAN class=mo id=id7018409>−</SPAN><SPAN class=mi1>m</SPAN><SPAN class=mo id=id7018417>)</SPAN><SPAN class=mo id=id7018421>)</SPAN><SPAN style="COLOR: red"></apply></SPAN></SPAN><SUB><SPAN class=mi1>N</SPAN></SUB></SPAN></SPAN><SPAN id=id7018386R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7018386M.offsetHeight; mrowStretch(id7018386L,'é','ê','ê','ë'); mrowStretch(id7018386R,'ù','ú','ú','û');</SCRIPT> </FONT></SPAN></SPAN></SPAN></LI></UL></DIV> <TABLE class=figure id=fig5 cellSpacing=0 cellPadding=0 width="50%" align=center border=0> <TBODY> <TR> <TD> <DIV class=inner-figure><FONT size=2><IMG class=media src= http://www.wikilearning.com/imagescc/20470/cconv_p4.png∞ ></FONT></DIV> <DIV class=caption><SPAN class=caption-before><FONT size=2>Figure 5</FONT></SPAN></DIV></TD></TR></TBODY></TABLE> <DIV class=para id=paraex4><FONT size=2>Multiplique <SPAN class=mrow id=id7018534><SPAN class=mrow id=id7018535><SPAN class=mrow id=id7018545><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7018548></SPAN><SPAN id=id7018551L>[</SPAN><SPAN id=id7018551M><SPAN class=mi1>m</SPAN></SPAN><SPAN id=id7018551R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7018551M.offsetHeight; mrowStretch(id7018551L,'é','ê','ê','ë'); mrowStretch(id7018551R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>y <SPAN class=mrow id=id7018579><SPAN class=mrow id=id7018580><SPAN class=mi>sume</SPAN></SPAN></SPAN> para dar: <SPAN class=mrow id=id7018601><SPAN class=mrow id=id7018606><SPAN class=mrow id=id7018617><SPAN class=mrow id=id7018618><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7018621></SPAN><SPAN id=id7018623L>[</SPAN><SPAN id=id7018623M><SPAN class=mn>1</SPAN></SPAN><SPAN id=id7018623R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7018623M.offsetHeight; mrowStretch(id7018623L,'é','ê','ê','ë'); mrowStretch(id7018623R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7018630>=</SPAN><SPAN class=mn>5</SPAN></SPAN></SPAN></SPAN> </FONT></DIV> <DIV class=para id=paraex5> <UL class=list id=eg_l3> <LI class=item><SPAN class=mrow id=id7018678><SPAN class=mrow id=id7018681><SPAN class=mrow id=id7018692><FONT size=2><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7018696></SPAN><SPAN id=id7018699L>[</SPAN><SPAN id=id7018699M><SPAN class=mrow id=id7018704><SPAN><SPAN style="COLOR: red"><apply></SPAN><SPAN class=mo id=id7018710>(</SPAN><SPAN class=mo id=id7018714>(</SPAN><SPAN class=mn>2</SPAN><SPAN class=mo id=id7018722>−</SPAN><SPAN class=mi1>m</SPAN><SPAN class=mo id=id7018730>)</SPAN><SPAN class=mo id=id7018734>)</SPAN><SPAN style="COLOR: red"></apply></SPAN></SPAN><SUB><SPAN class=mi1>N</SPAN></SUB></SPAN></SPAN><SPAN id=id7018699R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7018699M.offsetHeight; mrowStretch(id7018699L,'é','ê','ê','ë'); mrowStretch(id7018699R,'ù','ú','ú','û');</SCRIPT> </FONT></SPAN></SPAN></SPAN></LI></UL></DIV> <TABLE class=figure id=fig6 cellSpacing=0 cellPadding=0 width="50%" align=center border=0> <TBODY> <TR> <TD> <DIV class=inner-figure><FONT size=2><IMG class=media src= http://www.wikilearning.com/imagescc/20470/cconv_p5.png∞ ></FONT></DIV> <DIV class=caption><SPAN class=caption-before><FONT size=2>Figure 6</FONT></SPAN></DIV></TD></TR></TBODY></TABLE> <DIV class=para id=paraex6><FONT size=2>Multiplique <SPAN class=mrow id=id7018847><SPAN class=mrow id=id7018849><SPAN class=mrow id=id7018858><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7018861></SPAN><SPAN id=id7018865L>[</SPAN><SPAN id=id7018865M><SPAN class=mi1>m</SPAN></SPAN><SPAN id=id7018865R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7018865M.offsetHeight; mrowStretch(id7018865L,'é','ê','ê','ë'); mrowStretch(id7018865R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>y <SPAN class=mrow id=id7018892><SPAN class=mrow id=id7018894><SPAN class=mi>sume</SPAN></SPAN></SPAN> para dar: <SPAN class=mrow id=id7018914><SPAN class=mrow id=id7018919><SPAN class=mrow id=id7018930><SPAN class=mrow id=id7018931><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7018934></SPAN><SPAN id=id7018936L>[</SPAN><SPAN id=id7018936M><SPAN class=mn>2</SPAN></SPAN><SPAN id=id7018936R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7018936M.offsetHeight; mrowStretch(id7018936L,'é','ê','ê','ë'); mrowStretch(id7018936R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7018943>=</SPAN><SPAN class=mn>3</SPAN></SPAN></SPAN></SPAN> </FONT></DIV> <DIV class=para id=paraex7> <UL class=list id=eg_l4> <LI class=item><SPAN class=mrow id=id7018991><SPAN class=mrow id=id7018995><SPAN class=mrow id=id7019006><FONT size=2><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7019009></SPAN><SPAN id=id7019012L>[</SPAN><SPAN id=id7019012M><SPAN class=mrow id=id7019017><SPAN><SPAN style="COLOR: red"><apply></SPAN><SPAN class=mo id=id7019023>(</SPAN><SPAN class=mo id=id7019027>(</SPAN><SPAN class=mn>3</SPAN><SPAN class=mo id=id7019035>−</SPAN><SPAN class=mi1>m</SPAN><SPAN class=mo id=id7019000>)</SPAN><SPAN class=mo id=id7019043>)</SPAN><SPAN style="COLOR: red"></apply></SPAN></SPAN><SUB><SPAN class=mi1>N</SPAN></SUB></SPAN></SPAN><SPAN id=id7019012R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7019012M.offsetHeight; mrowStretch(id7019012L,'é','ê','ê','ë'); mrowStretch(id7019012R,'ù','ú','ú','û');</SCRIPT> </FONT></SPAN></SPAN></SPAN></LI></UL></DIV> <TABLE class=figure id=fig7 cellSpacing=0 cellPadding=0 width="50%" align=center border=0> <TBODY> <TR> <TD> <DIV class=inner-figure><FONT size=2><IMG class=media src= http://www.wikilearning.com/imagescc/20470/cconv_p6.png∞ ></FONT></DIV> <DIV class=caption><SPAN class=caption-before><FONT size=2>Figure 7</FONT></SPAN></DIV></TD></TR></TBODY></TABLE> <DIV class=para id=paraex8><FONT size=2>Multiplique <SPAN class=mrow id=id7019158><SPAN class=mrow id=id7019161><SPAN class=mrow id=id7019170><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7019173></SPAN><SPAN id=id7019178L>[</SPAN><SPAN id=id7019178M><SPAN class=mi1>m</SPAN></SPAN><SPAN id=id7019178R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7019178M.offsetHeight; mrowStretch(id7019178L,'é','ê','ê','ë'); mrowStretch(id7019178R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>y <SPAN class=mrow id=id7019205><SPAN class=mrow id=id7019166><SPAN class=mi>sume</SPAN></SPAN></SPAN> para dar: <SPAN class=mrow id=id7019221><SPAN class=mrow id=id7019229><SPAN class=mrow id=id7019243><SPAN class=mrow id=id7019244><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7019248></SPAN><SPAN id=id7019250L>[</SPAN><SPAN id=id7019250M><SPAN class=mn>3</SPAN></SPAN><SPAN id=id7019250R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7019250M.offsetHeight; mrowStretch(id7019250L,'é','ê','ê','ë'); mrowStretch(id7019250R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7019257>=</SPAN><SPAN class=mn>1</SPAN></SPAN></SPAN></SPAN> </FONT></DIV></DIV> <DIV class=example id=vi_demo><FONT size=2><SPAN class=example-before>Example 2</SPAN><SPAN class=example-name> </SPAN> </FONT> <DIV class=para id=para_vi_demo_1><FONT size=2>La Siguiente Demostración le permite este algoritmo. Vea </FONT><FONT size=2>aquí</FONT><FONT size=2> para instrucciones de como se usa este demo. </FONT></DIV> <DIV class="media labview example"><FONT size=2><SPAN class=example-before>LabVIEW Example: </SPAN>(</FONT><FONT size=2>run</FONT><FONT size=2>) (</FONT><FONT size=2>source</FONT><FONT size=2>) </FONT></DIV></DIV></DIV> <DIV class=section id=sub2> <H2><FONT size=2>Algoritmo Alterno</FONT></H2> <UL class=list id=altmeth><FONT size=2><SPAN class=name>Algoritmo de Convolución Circular Alterno</SPAN> </FONT> <LI class=item><FONT size=2>Paso 1: Calcule el DFT de <SPAN class=mrow id=id7019380><SPAN class=mrow id=id7019382><SPAN class=mrow id=id7019392><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7019395></SPAN><SPAN id=id7019398L>[</SPAN><SPAN id=id7019398M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7019398R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7019398M.offsetHeight; mrowStretch(id7019398L,'é','ê','ê','ë'); mrowStretch(id7019398R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>que da <SPAN class=mrow id=id7019426><SPAN class=mrow id=id7019430><SPAN class=mrow id=id7019435><SPAN class=mi1>F</SPAN><SPAN class=mo id=id7019439></SPAN><SPAN id=id7019442L>[</SPAN><SPAN id=id7019442M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7019442R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7019442M.offsetHeight; mrowStretch(id7019442L,'é','ê','ê','ë'); mrowStretch(id7019442R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>y calcule el DFT de <SPAN class=mrow id=id7019470><SPAN class=mrow id=id7019434><SPAN class=mrow id=id7019480><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7019483></SPAN><SPAN id=id7019486L>[</SPAN><SPAN id=id7019486M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7019486R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7019486M.offsetHeight; mrowStretch(id7019486L,'é','ê','ê','ë'); mrowStretch(id7019486R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>que da <SPAN class=mrow id=id7019514><SPAN class=mrow id=id7019515><SPAN class=mrow id=id7019522><SPAN class=mi1>H</SPAN><SPAN class=mo id=id7019526></SPAN><SPAN id=id7019529L>[</SPAN><SPAN id=id7019529M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7019529R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7019529M.offsetHeight; mrowStretch(id7019529L,'é','ê','ê','ë'); mrowStretch(id7019529R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>. </FONT> <LI class=item><FONT size=2>Paso 2: Multiplique punto por punto <SPAN class=mrow id=id7019562><SPAN class=mrow id=id7019563><SPAN class=mrow id=id7019577><SPAN class=mrow id=id7019578><SPAN class=mi1>Y</SPAN><SPAN class=mo id=id7019581></SPAN><SPAN id=id7019583L>[</SPAN><SPAN id=id7019583M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7019583R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7019583M.offsetHeight; mrowStretch(id7019583L,'é','ê','ê','ë'); mrowStretch(id7019583R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7019591>=</SPAN><SPAN class=mrow id=id7019594><SPAN class=mrow id=id7019598><SPAN class=mi1>F</SPAN><SPAN class=mo id=id7019601></SPAN><SPAN id=id7019604L>[</SPAN><SPAN id=id7019604M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7019604R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7019604M.offsetHeight; mrowStretch(id7019604L,'é','ê','ê','ë'); mrowStretch(id7019604R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7019611></SPAN><SPAN class=mrow id=id7019614><SPAN class=mi1>H</SPAN><SPAN class=mo id=id7019617></SPAN><SPAN id=id7019619L>[</SPAN><SPAN id=id7019619M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7019619R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7019619M.offsetHeight; mrowStretch(id7019619L,'é','ê','ê','ë'); mrowStretch(id7019619R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN></SPAN></SPAN></FONT> <LI class=item><FONT size=2>Paso 3: Invierta el DFT <SPAN class=mrow id=id7019697><SPAN class=mrow id=id7019575><SPAN class=mrow id=id7019700><SPAN class=mi1>Y</SPAN><SPAN class=mo id=id7019705></SPAN><SPAN id=id7019707L>[</SPAN><SPAN id=id7019707M><SPAN class=mi1>k</SPAN></SPAN><SPAN id=id7019707R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7019707M.offsetHeight; mrowStretch(id7019707L,'é','ê','ê','ë'); mrowStretch(id7019707R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN>que da <SPAN class=mrow id=id7019735><SPAN class=mrow id=id7019739><SPAN class=mrow id=id7019744><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7019748></SPAN><SPAN id=id7019750L>[</SPAN><SPAN id=id7019750M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7019750R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7019750M.offsetHeight; mrowStretch(id7019750L,'é','ê','ê','ë'); mrowStretch(id7019750R,'ù','ú','ú','û');</SCRIPT> </SPAN></SPAN></SPAN></FONT></LI></UL> <DIV class=para id=palt1><FONT size=2>Parece una manera repetitiva de hacer las cosas, <I>pero</I> existen maneras <I>rápidas</I> de calcular una secuencia DFT. </FONT></DIV> <DIV class=para id=palt2><FONT size=2>Para convolucionar circularmente dos secuencias de <SPAN class=mrow id=id7019801><SPAN class=mrow id=id7019806><SPAN class=mn>2</SPAN></SPAN></SPAN> <SPAN class=mrow id=id7019818><SPAN class=mrow id=id7019821><SPAN class=mi1>N</SPAN></SPAN></SPAN>-puntos: </FONT><SPAN class=mrow id=id7019836><SPAN class=mrow id=id7019840><SPAN class=mrow id=id7019856><FONT size=2><SPAN class=mrow id=id7019857><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7019860></SPAN><SPAN id=id7019862L>[</SPAN><SPAN id=id7019862M><SPAN class=mi1>n</SPAN></SPAN><SPAN id=id7019862R>]</SPAN> <SCRIPT type=text/javascript> var mrowH = id7019862M.offsetHeight; mrowStretch(id7019862L,'é','ê','ê','ë'); mrowStretch(id7019862R,'ù','ú','ú','û');</SCRIPT> </SPAN><SPAN class=mo id=id7019870>=</SPAN></FONT><SPAN class=mrow id=id7019873><FONT size=2> </FONT> <TABLE class=munderover> <TBODY> <TR> <TD><SPAN class=mrow id=id7019889><SPAN class=mrow id=id7019894><FONT size=2><SPAN class=mi1>N</SPAN><SPAN class=mo id=id7019897>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR> <TR> <TD><SPAN class=mo id=id7019876><FONT size=2>∑</FONT></SPAN